Semigroups with length morphisms

Abstract

The class of metrical semigroups is defined as the set consisting of those semigroups which can be homomorphically mapped into the semigroup of natural numbers (without zero) under addition. The finitely generated members of this class are characterised and the infinitely generated case is discussed. A semigroup is called locally metrical if every finitely generated subsemigroup is metrical. The classical Green's relations are trivial on any metrical semigroup. Generalisations 𝓗+, 𝓛+ and 𝓡+ of the Green's relations are defined and it is shown that for any cancellative metrical semigroup, S, 𝓗 + is " as big as possible " if and only if S is isomorphic to a special type of semidirect product of 𝗡 and a group. Lyndon's characterisation of free groups by length functions is discussed andalink between length functions, metrical semigroups and semigroups embeddable into free semigroups is investigated. Next the maximal locally metrical ideal of a semigroup is discussed, and the class of t-compressible semigroups is defined as the set consisting of those semigroups that have no locally metrical ideal. The class of t-compressible semigroups is seen to contain the classes of regular and simple semigroups. Finally it is shown that a large class of semigroups can be decomposed into a chain of locally metrical ideals together with a t-compressible semigroup.